Let \(\mathcal {P}=\lbrace p_{\Sigma }:\Sigma \in \mathcal {M}^+\rbrace \)  be the space of all densities \(p_{\Sigma }\)  and consider a sieve \(\mathcal {P}_n=\lbrace p_{\Sigma }:\Sigma \in \mathcal {M}^+,s(\Sigma ,\delta _n)\le s_n,1/\tau \le \lambda _{\min }(\Sigma )\le \lambda _{\max }(\Sigma )\le \tau ,||\Sigma ||_{\infty }\le L_n\rbrace \subset \mathcal {P}\) , where \(\delta _n,s_n,\)  and \(L_n\)  are to be determined in Theorem REF . Denote the \(\epsilon -\)  packing number for subset \(\mathcal {A}\)  of metric space \((\mathcal {S},d)\)  by \(\text{D}(\epsilon ,\mathcal {A},d)\)  , i.e., \(\text{D}(\epsilon ,\mathcal {A},d)\)  is the minimum number of \(d\) -balls of size \(\epsilon \)  in \(\mathcal {S}\)  needed to cover \(\mathcal {A}\)  under metric \(d\) . Now, under regular conditions, we prove Theorem REF  by verifying conditions
(10)-(12) of Lemma 5.1 in [1]}, which is a version of Theorem 2.1 in [2]}. Define a function \(s(\cdot ,\cdot )\)  on \(\mathcal {M}^+\times \mathbb {R}^+\)  by \(s(\mathcal {K},\delta )=\sum _{i<j}{1}(|k_{ij}|\ge \delta )\) , where \(\mathcal {K}=(k_{ij})\) . Note that \(s\)  is equivalent to the number of edges in the graph induced by \(\mathcal {K}\)  with a threshold \(\delta \) . Define the set
\(\mathcal {U}(\delta _n,s_n,L_n,\tau )=\lbrace \Sigma \in \mathcal {M}^+:s(\Sigma ,\delta _n)\le s_n,1/\tau \le \lambda _{\min }(\Sigma )\le \lambda _{\max }(\Sigma )\le \tau ,||\Sigma ||_{\infty }\le L_n\rbrace .\) 
